Elongated triangular bipyramid

Short description: 14th Johnson solid; triangular prism capped with tetrahedra

Elongated triangular bipyramid

Type| Johnson
J13 – J14 – J15
Faces| 6 triangles
3 squares
Edges| 15
Vertices| 8
Vertex configuration| 2(33)
6(32.42)
Symmetry group| D3h, [3,2], (*322)
Rotation group| D3, [3,2]+, (322)
Dual polyhedron| Triangular bifrustum
Properties| convex
Net

File:3D Johnson J14.stl

In geometry, the elongated triangular bipyramid (or dipyramid) or triakis triangular prism is one of the Johnson solids (J14), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid (J12) by inserting a triangular prism between its congruent halves.

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.[2]

## Contents

* 1 Formulae
* 1.1 Dual polyhedron
* 2 References
* 3 External links

## Formulae

The following formulae for volume ([math]\displaystyle{ V }[/math]), surface area ([math]\displaystyle{ A }[/math]) and height ([math]\displaystyle{ H }[/math]) can be used if all faces are regular, with edge length a:

[math]\displaystyle{ V=\left(\frac{1}{12}\left(2\sqrt{2}+3\sqrt{3}\right)\right)\cdot a^3\approx 0.668715...a^3 }[/math][3][4]
[math]\displaystyle{ A=\left(\frac{3}{2}\left(2+\sqrt{3}\right)\right)\cdot a^2\approx 5.59808...a^2 }[/math][3][4]
[math]\displaystyle{ H = \frac{3 + 2\sqrt{6}}{3}\cdot a \approx \cdot 2.63299...a }[/math][4]

### Dual polyhedron

The dual of the elongated triangular bipyramid is called a triangular bifrustum and has 8 faces: 6 trapezoidal and 2 triangular.

Dual elongated triangular bipyramid  | Net of dual

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## References

1. ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8 .
2. ↑ Gerdes, Paulus (2009), "Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education", ZDM Mathematics Education 42 (1): 11–17, doi:10.1007/s11858-009-0208-2 .
3. ↑ 3.0 3.1 Stephen Wolfram, "Elongated triangular dipyramid" from Wolfram Alpha. Retrieved July 22, 2010.
4. ↑ 4.0 4.1 4.2 Sapiña, R.. "Area and volume of the Johnson solid J14" (in es). Problemas y Ecuaciones. ISSN 2659-9899. https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J14/calculadora-area-volumen-formulas.html. Retrieved 2020-09-04.

## External links

* Eric W. Weisstein, Elongated triangular bipyramid (Johnson solid) at MathWorld.

* v
* t
* e

Johnson solids

Pyramids, cupolae and rotundae|

* square pyramid
* pentagonal pyramid
* triangular cupola
* square cupola
* pentagonal cupola
* pentagonal rotunda

Modified pyramids|

* elongated triangular pyramid
* elongated square pyramid
* elongated pentagonal pyramid
* gyroelongated square pyramid
* gyroelongated pentagonal pyramid
* triangular bipyramid
* square bipyramid
* pentagonal bipyramid
* elongated triangular bipyramid
* elongated square bipyramid
* elongated pentagonal bipyramid
* gyroelongated square bipyramid

Modified cupolae and rotundae|

* elongated triangular cupola
* elongated square cupola
* elongated pentagonal cupola
* elongated pentagonal rotunda
* gyroelongated triangular cupola
* gyroelongated square cupola
* gyroelongated pentagonal cupola
* gyroelongated pentagonal rotunda
* gyrobifastigium
* triangular orthobicupola
* square orthobicupola
* square gyrobicupola
* pentagonal orthobicupola
* pentagonal gyrobicupola
* pentagonal orthocupolarotunda
* pentagonal gyrocupolarotunda
* pentagonal orthobirotunda
* elongated triangular orthobicupola
* elongated triangular gyrobicupola
* elongated square gyrobicupola
* elongated pentagonal orthobicupola
* elongated pentagonal gyrobicupola
* elongated pentagonal orthocupolarotunda
* elongated pentagonal gyrocupolarotunda
* elongated pentagonal orthobirotunda
* elongated pentagonal gyrobirotunda
* gyroelongated triangular bicupola
* gyroelongated square bicupola
* gyroelongated pentagonal bicupola
* gyroelongated pentagonal cupolarotunda
* gyroelongated pentagonal birotunda

Augmented prisms|

* augmented triangular prism
* biaugmented triangular prism
* triaugmented triangular prism
* augmented pentagonal prism
* biaugmented pentagonal prism
* augmented hexagonal prism
* parabiaugmented hexagonal prism
* metabiaugmented hexagonal prism
* triaugmented hexagonal prism

Modified Platonic solids|

* augmented dodecahedron
* parabiaugmented dodecahedron
* metabiaugmented dodecahedron
* triaugmented dodecahedron
* metabidiminished icosahedron
* tridiminished icosahedron
* augmented tridiminished icosahedron

Modified Archimedean solids|

* augmented truncated tetrahedron
* augmented truncated cube
* biaugmented truncated cube
* augmented truncated dodecahedron
* parabiaugmented truncated dodecahedron
* metabiaugmented truncated dodecahedron
* triaugmented truncated dodecahedron
* gyrate rhombicosidodecahedron
* parabigyrate rhombicosidodecahedron
* metabigyrate rhombicosidodecahedron
* trigyrate rhombicosidodecahedron
* diminished rhombicosidodecahedron
* paragyrate diminished rhombicosidodecahedron
* metagyrate diminished rhombicosidodecahedron
* bigyrate diminished rhombicosidodecahedron
* parabidiminished rhombicosidodecahedron
* metabidiminished rhombicosidodecahedron
* gyrate bidiminished rhombicosidodecahedron
* tridiminished rhombicosidodecahedron

Elementary solids|

* snub disphenoid
* snub square antiprism
* sphenocorona
* augmented sphenocorona
* sphenomegacorona
* hebesphenomegacorona
* disphenocingulum
* bilunabirotunda
* triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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